1

Affine Neural Network Based Predictive ControlApplied to a Distributed Solar Collector Field

Paulo Gil, Member, IEEE, Jorge Henriques, Alberto Cardoso, Paulo Carvalho,and Antonio Dourado, Member, IEEE

Abstract—This paper presents experimental results concerningthe control of a distributed solar collector field, where the mainobjective concerns the regulation of the outlet oil temperature bysuitably manipulating the oil flow rate. This is achieved by meansof a constrained nonlinear adaptive model-based predictivecontrol framework where the control action sequence is obtainedby solving an open loop optimization problem, subject to a setof constraints. The plant dynamics is approximated by an affinestate-space neural network, whose complexity is specified in termsof the cardinality of dominant singular values associated witha subspace oblique projection of data driven Hankel matrices.The neural network is first trained offline and subsequentlyimproved through a recursive updating of its weights and biases,based on a dual unscented Kalman filter. The control schemewas implemented on the Acurex field of the Plataforma Solarde Almerıa, Spain. Results from these experiments demonstratethe feasibility of the proposed framework, and highlighted theability to cope with time-varying and unmodelled dynamics,under the form of disturbances, and its inherent capability foraccommodating actuation faults.

Index Terms—Model-based predictive control, adaptive con-trol, constrained optimization, affine state-space neural networks,online training, unscented Kalman filter, distributed solar collec-tor field.

I. INTRODUCTION

THE modern societies way of life has been proppedup on a high pace of fossil energy consumption per

capita, which was constantly growing over the last century [1].Inevitably, this behaviour is prompting a depletion of fossilenergy reserves that is accompanied by an increase of airpollution, along with other forms of environmental degradation[2]. In order to mitigate such impacts on the environment,without compromising economic growth, it is crucial to im-plement a world-wide paradigm shift towards a non-carbonbased economy. It implies not only energy conservation, butalso the increase of renewable energy sources mix, such as,wind, hydroelectric or solar energy, which currently counts forless than 10 %, at world scale1.

In the case of thermal solar power for electricity generation,there are some commercial systems in operation, most based

P. Gil is with the Departmento de Engenharia Electrotecnica, Faculdadede Ciencias e Tecnologia, Universidade Nova de Lisboa, Caparica, 2829-516,Portugal, e-mail: psg@fct.unl.pt, and with Centre for Informatics and Systemsof the University of Coimbra (CISUC), University of Coimbra, Polo II, 3030-290, Coimbra, Portugal.

J. Henriques, A. Cardoso, P. Carvalho and A. Dourado are with CISUC,Informatics Engineering Department, University of Coimbra, Polo II, 3030-290, Coimbra, Portugal.

Manuscript received on July 19, 20121http://ourenergyfutures.org.

on parabolic trough solar collectors, namely, the Andasol I& II in the Plateau of Guadix, Granada, Spain and the SolarEnergy Generating Systems, in the Mojave Desert, California,USA (see e.g. [3] and references therein).

Parabolic trough solar systems concentrate irradiation ona linear absorber tube located at the focal line, while a heattransfer fluid, typically a synthetic oil, flows along the receiverpipes. In order to maximize the collected energy, regardless ofpossible disturbances acting on the system, namely, changes ondirect solar radiation due to cloud cover, inlet oil temperaturedrift, or optical efficiency degradation, due to dirt depositionon the mirrors’ surface, it is crucial that the control systemshould be designed in terms of favouring the closed loopperformance and robustness alike [4].

Among advanced control techniques applied to distributedsolar collector fields (DSC) to control the outlet heatingfluid’s temperature (see e.g. [5]) model-based predictive con-trol (MPC) techniques have a number of appealing features,namely, providing optimal or suboptimal sequences of controlactions, the potential for dealing with both input and outputconstraints, and the inherent ability for designing robust con-strained control systems.

Different predictive control schemes have been applied toDSC systems (see e.g. [6]–[8]). Essentially, they all consideran explicit dynamic model of the plant within an optimisationthread in order to come up with an optimised sequence ofcontrol actions. In addition, most of the implementations alsoinclude a feedforward compensator in the control system tocope with measurable disturbances [6]. Although few imple-mentation based on fixed parameters can be found in theliterature, mainly because of the inherent nonlinear time-varying plant dynamics, it is worth mentioning the work ofCamacho [9] where a generalized predictive controller wasimplemented on the Acurex field of the Plataforma Solar deAlmerıa (PSA). In Camacho and Berenguel [6] an adaptiverobust predictive control scheme was derived based on asimplified transfer function of the plant, considering the poleslocation fixed and zeros robustly identified online. Accordingto the authors this control scheme showed poor performancein terms of static error, in the case of radiation and inletoil temperature disturbances. Coito et al. [8] and Silva et al.[10] implemented a self-tuning control by multistep horizons(MUSMAR) framework on the Acurex field, while in [11]the authors propose a MUSMAR variant based on a bi-criteria optimization in order to improve the control system’sperformance in start-up transients. Experimental results havedemonstrated the feasibility of these receding horizon ap-

2

proaches. In Pickhardt and Silva [12] a nonlinear predictivecontrol scheme is proposed assuming an input-output modelof the plant within a receding horizon framework. The model’sparameters are updated online at every sampling time to allowaccommodating time-varying effects and modelling errors.Although the obtained results were promising, they show asignificant underperformance of the control system in thepresence of mild to severe radiation disturbances. In addition,as the authors stress, the underlying control law cannot in-corporate an integral effect and the computation overhead inthe optimization stage is quite dependent on the predictionhorizon, which may have a direct impact on the samplingfrequency.

A drawback of considering true nonlinear model-basedpredictive control (NMPC) techniques is linked to the needof solving a non-convex optimisation problem. Specifically,the convergence to a feasible/optimal solution and stabilityconditions cannot be guaranteed in advance, and the under-lying computation time might exceed, by far, the specifiedsampling period. Moreover, in many cases, finding a goodanalytical model of the plant might be a very demanding,time consuming and costly task. As such, system identificationshould be favoured against analytical modelling, based on phe-nomenological considerations. An example of nonlinear blackbox structures are the artificial neural networks, which arequite appealing in terms of approximation and generalisationcapabilities (see e.g. [13]–[17].

The above discussion motivates the development of ageneric conceptual platform for nonlinear control system de-sign, relying on a model-based predictive control framework,with the model of the plant described by an affine state-space three-layered neural network. The number of neuronsto be included in the hidden layer is selected taking intoaccount the specificity of the model structure and by applyingthe singular value decomposition (SVD) to a given obliquesubspace projection. In order to cope with time-varying dy-namics and modelling errors, an indirect adaptive scheme issuggested, where the neural network weights are updated bya recursive updating technique based on a dual unscentedKalman filter (DUKF) [18]. Finally, it should be mentionedthat any comparisons, either in terms of performance orcomplexity, against other control techniques are out of scopein this work, being the reader referred to [4] and [5], andreferences therein, for a comprehensive discussion on controlschemes applied to distributed solar collector fields.

The remainder of this paper is organized as follows. Sec-tion II introduces the affine state-space neural network topol-ogy, and describes the approaches considered for managing itscomplexity and training methodologies. Section III formulatesthe generic model-based control (MPC) scheme. In Section IVthe distributed solar collector field (Acurex) of the PlataformaSolar de Almerıa, Spain, is described, while in Section Vsome experimental results are presented and discussed. Finally,Section VI concludes the paper.

II. AFFINE STATE-SPACE NEURAL NETWORKS

Modelling nonlinear dynamic systems has been the maindriving force for developing robust black-box topologies (see

e.g. [19], [20]) in order to deal with unknown dynamics,nonlinearities and uncertainties. In this context, the ability ofneural networks to learn based on input-output data and gener-alize to unseen data are quite appealing, in particular when theprocess dynamics is highly complex. In addition, their intrinsicapproximation capabilities, scalability and inherent structuralflexibility to cope with a larger class of problems are invaluablecharacteristics.

In the case of three-layered feedforward neural networks(input layer - hidden layer - output layer) consisting ofsufficient number of hidden-layer’s neurons with sigmoidalactivation function, they can, in theory, approximate, to anylevel of accuracy, a given continuous nonlinear function (seee.g. [21]–[24]). However, this neural network topology is staticby nature and consequently cannot inherently approximatespatio-temporal information. The way these structures canbe used to emulate nonlinear systems’ dynamics is typicallycarried out by providing the network with a regressor vectorcomprising past inputs and past outputs, being these structuresdubbed as NARX (Nonlinear AutoRegressive with eXogenousinputs) neural networks.

Although NARX neural networks are effective in emulatingnonlinear systems’ dynamics (see e.g. [25]) they have someinescapable drawbacks. Specifically, they can only encode afinite number of previous inputs and outputs, they suffer fromhigh sensitivity to the regressor’s lag windows, and show anotorious susceptibility to noisy data.

A conceptually different way to represent nonlinear systemdynamics is by incorporating feedback connections within thehidden layer of a three-layered neural network, where theremaining two layers are represented by the input layer andoutput layer. This architecture, referred to as state-space neuralnetwork, represents a broader class of nonlinear dynamicsystems, and not only is less vulnerable to noisy data, butalso the whole system history can inherently be embeddedinto the model. Moreover, state-space neural networks provideuniversal identification models in the restricted sense that theycan approximate uniformly any multi input - multi outputnonlinear dynamics over a finite time horizon, for everycontinuous and bounded input signal (see e.g. [26]).

A. Neural network architecture

The affine discrete-time state-space neural network topologyconsidered in this work consists of three layers, as depictedin Fig. 1. The input layer and the output layer have as manyneurons as the number of exogenous inputs and outputs of thedynamic system to be modeled, while the number of neuronsto be incorporated within the hidden layer should be judi-ciously selected for the sake of generalization performance.A neural network comprising a deficient number of hiddenlayer neurons may not be feasible to appropriately emulate theunderlying system dynamics, while a number of neurons largerthan theoretically required may result in limited generalizationperformance due to overfitting. This problem is addressed inSection II-B.

The activation functions of neurons included in the hiddenlayer are both linear and sigmoidal, since in conjunction they

3

y(k+1)

u(k)

D C

B

A

x(k)

q-1

x(k+1)

E

Fig. 1. Affine state-space neural network.

improve the neural predictor performance in case of mildnonlinear dynamics [27], and contribute to a less demandingtraining phase.

In state-space form the corresponding analytical model canbe written as,

x (k + 1) = Ax (k) +B u (k) +Dσ (E x (k))

y (k) = C x (k)(1)

where x ∈ Rn denotes the hyperstate space vector, y ∈ Rp theoutput vector and u ∈ Rm the input vector. Matrices A, B, C,D and E are real-valued matrices of appropriate dimensions,while the nonlinear activation function σ(·) is a continuous anddifferentiable sigmoidal function, upper and lower bounded,satisfying the following conditions [28]:• limt→±∞σ (t) = ±1;• σ (t) = 0⇐ t = 0;• σ′ (t) > 0;• limt→±∞σ

′ (t) = 0;• max (σ′ (t)) ≤ 1⇐ t = 0.The reader is referred to [29] for a comprehensive survey

on activation functions in artificial neural networks and [30] toa thorough study on stability conditions applied to the affinestate-space architecture considered in this work.

B. Neural network complexity

This section deals with the problem of selecting an effectivenumber of hidden layer neurons for the system represented inFig. 1. Formally, it can be tackled by estimating an effectiveorder for the affine state-space model (Eq. 1), assuming alinear approximation to the system’s dynamics [31].

Let us assume a generic finite dimensional discrete-timeinvariant linear system represented in state-space form as,

x (k + 1) = Ax (k) +B u (k) + η (k)

y (k) = C x (k) +Du (k) + ϑ (k)(2)

where x ∈ Rn, y ∈ Rp, u ∈ Rm, A ∈ Rn×n, B ∈ Rn×m,C ∈ Rp×n and D ∈ Rp×m. Vectors ϑ ∈ Rp and η ∈ Rn

are, respectively, unobserved Gaussian distributed, zero mean,white noise sequences accounting for the measurement noiseand process noise. Additionally, suppose that the available datacollected from the plant are ergodic and go to infinity, andEq. (2) satisfies the orthogonality property, that is,

E[(x (k)u (k)

)(ηT (k) ϑT (k)

)]= 0 (3)

with E denoting the expectation.Consider now the past and future input block Hankel

matrices, respectively, Up ≡ U0|i−1 and Uf ≡ Ui|2i−1 givenby:

Up =

u (0) u (1) · · · u (j − 1)u (1) u (2) · · · u (j)

......

. . ....

u (i− 1) u (i) · · · u (i+ j − 2)

(4)

Uf =

u (i) u (i+ 1) · · · u (i+ j − 1)

u (i+ 1) u (i+ 2) · · · u (i+ j)...

.... . .

...u (2i− 1) u (2i) · · · u (2i+ j − 2)

(5)

and the past and future output block Hankel matrices, Yp ≡Y0|i−1 and Yf ≡ Yi|2i−1 given according to,

Yp =

y (0) y (1) · · · y (j − 1)y (1) y (2) · · · y (j)

......

. . ....

y (i− 1) y (i) · · · y (i+ j − 2)

(6)

Yf =

y (i) y (i+ 1) · · · y (i+ j − 1)

y (i+ 1) y (i+ 2) · · · y (i+ j)...

.... . .

...y (2i− 1) y (2i) · · · y (2i+ j − 2)

(7)

where the number of block rows i should be selected largerthan the expected maximum order of the system under iden-tification, that is, n i and the number of block columnsj →∞.

The effective order for the linear model (2), which dependson the information embedded in the data set, can be retrievedfrom the rank of a subspace projection Π involving the aboveHankel matrices (see e.g. [32].)

Definition 1 (Oblique projection). The oblique projection ofthe row space of P ∈ Rp×j along the row space of Υ ∈ Rq×j

on the row space of Γ ∈ Rl×j is given by:

P/ΥΓ = P[ΓT ΥT

] [(ΓΓT ΓΥT

ΥΓT ΥΥT

)†]1:l

Γ (8)

withM† denoting the Moore-Penrose pseudoinverse of matrixM.

Theorem 1 (Main subspace identification theorem [33]).Under the assumptions that: a) the deterministic input isuncorrelated with the process noise η and measurement noiseϑ; b) the process noise η and the measurement noise ϑare not identically zero; c) the exogenous input sequence ispersistently exciting [34] of order 2 i; d) the data set is large(j →∞), then:

4

1) The projection Πi can be defined as the oblique projec-tion of the row space of Yf on the past input/output rowspace along the row space of Uf,

Πi = YfUf

(Up

Yp

)(9)

2) The order of the linear discrete-time system is given bythe number of non-zero singular values of Πi.

The number of non-zero singular values of Πi revealswhether this projection is of full rank or rank deficient. Inorder to obtain the rank of Πi the singular value decomposition(SVD) can be applied to the above projection, under the strictconditions of Theorem 1. Since the SVD provides orthonormalbases for range and null spaces, the cardinality of the columnspace of Πi provides an estimate to the order of the linearsystem (2).

When this approach is applied to finite data sets (j ∞)or in the case of data collected from nonlinear systems, evenfor large data sets exempt from coloured noise, the obliqueprojection matrix Πi is full rank, that is, rank (Πi) = i · p,irrespective of the number of row blocks i of the correspondingblock Hankel matrices. In such cases it is imperative to reducethe column space of Πi, by taking into account, for instance,the dominant singular values cardinality.

In the MATLAB c© System Identification Toolbox [35]the implemented approach to estimate the system’s order rsearches for a gap in the singular values, being the order givenby the largest integer, so that the corresponding singular valueis greater than the geometric mean of the largest singular valueσ1 and the smallest non-zero singular value σs, that is,

r = max

r : log σr >

1

2(log σ1 + log σs)

(10)

A similar approach based on the search for a gap ispresented in [36], where the estimate model order is givenby,

r = max

r :

σrσr+1

>σ`σ`+1

,∀ ` = 1, . . . , s ∧ ` 6= r

(11)

C. Training

The affine state-space neural network training is first carriedout offline by using the Levenberg-Marquardt algorithm (seee.g. [37]), in particular because of its efficiency when thenetwork contains no more than a few hundred parameters[38]. In the case of indirect adaptive MPC schemes thisset of parameters is subsequently updated online by meansof a recursive approach based on a dual Kalman filter. Itsimplementation relies, essentially, on the propagation of meansand covariances through nonlinear transformations, referred toas unscented transformation (see e.g. [18], [39]). In the dualunscented Kalman filter framework both states and parame-ters are computed simultaneously in two different stages: i)the time update stage concerns the evaluation of one step-ahead predictions for states and parameters, while ii) in themeasurement update a correction is made to these estimateson the basis of current noisy measurements.

Assume the general state-space nonlinear system written as:

x (k + 1) = f(x (k) , u (k) , wf (k) , k

)+ η (k)

z (k) = g (x (k) , u (k) , wg (k) , k) + ϑ (k)(12)

where x ∈ Rn, u ∈ Rm, z ∈ Rp; f (·) and g (·) are realvector nonlinear functions with parametrisations wf and wg;η ∈ Rn and ϑ ∈ Rp are random variables representing theprocess noise and measurement noise, respectively.

Taking into account the nonlinear black-box structure de-scribed by Eq. (12) the corresponding DUKF equations are asfollows:

Weights estimationTime update:

Ωi (k|k − 1) = Ωi (k − 1|k − 1) (13)

Pw (k|k − 1) = Pw (k − 1|k − 1) + Ψ (14)

Zwi (k|k − 1) = g (x (k − 1|k − 1) , u (k − 1|k − 1)

,Ωi (k|k − 1) , k)(15)

zw (k|k − 1) =

2Nw∑l=0

Γwl Z

wl (k|k − 1) (16)

Pwz (k|k − 1) =

2Nw∑l=0

Γwl [Zw

l (k|k − 1)− zw (k|k − 1)]

· [Zwl (k|k − 1)− zw (k|k − 1)]

T

+ Υ

(17)

Pwz (k|k − 1) =

2Nw∑l=0

Γwl [Ωl (k|k − 1)− w (k|k − 1)]

· [Zwl (k|k − 1)− zw (k|k − 1)]

T

(18)Measurement update:

Kw (k) = Pwz (k|k − 1) [Pwz (k|k − 1)]

−1 (19)

w (k|k) = w (k|k − 1) +Kw (k) [z (k)− zw (k|k − 1)] (20)

Pw (k|k) = Pw (k|k − 1)−Kw (k)Pwz (k|k − 1)

· (Kw (k))T (21)

with Ω the sigma points matrices of the vectorized affinestate-space neural network weights w =

(wf |wg

), Pw, P

wz

and Pwz covariance matrices, Zw the observation matrix, zwthe weighted observation vector; Γ is the vector of weightingcoefficients, K the Kalman gain; Υ denotes the measurementnoise variance and Ψ the process noise.

State vector estimationTime update:

χl (k|k − 1) = f (χl (k − 1|k − 1) , u (k − 1)

w (k − 1|k − 1) , k)(22)

x (k|k − 1) =

2Nx∑l=0

Γxl · χl (k|k − 1) (23)

5

Px (k|k − 1) =

2Nx∑l=0

Γxl [χl (k|k − 1)− x (k|k − 1)]

· [χl (k|k − 1)− x (k|k − 1)]+ Θ

(24)

Zxi (k|k − 1) = g (χl (k − 1|k − 1) , u (k − 1|k − 1)

, w (k|k − 1) , k)(25)

zx (k|k − 1) =

2Nx∑l=0

Γxl Z

xl (k|k − 1) (26)

P xz (k|k − 1) =

2Nx∑l=0

Γxl [Zx

l (k|k − 1)− zx (k|k − 1)]

· [Zxl (k|k − 1)− zx (k|k − 1)]

T

+ Υ

(27)

Pxz (k|k − 1) =

2Nx∑l=0

Γxl [χl (k|k − 1)− x (k|k − 1)]

· [Zxl (k|k − 1)− zx (k|k − 1)]

T (28)

Measurement update:

Kx (k) = Pxz (k|k − 1) [P xz (k|k − 1)]

−1 (29)

x (k|k) = x (k|k − 1) +Kx (k) [z (k)− zx (k|k − 1)] (30)

Px (k|k) = Px (k|k − 1)−Kx (k)P xz (k|k − 1)

· (Kx (k))T (31)

where Θ is the process noise variance matrix and χ the sigmapoints matrices associated with the hyperstate vector x.

III. MODEL-BASED PREDICTIVE CONTROL

Model-based predictive control is a discrete-time techniquewhere an explicit dynamic model of the plant is used to predictthe system outputs over a finite horizon P , while a sequenceof control actions is adjusted during a finite control horizonM in order to minimize a given cost function (see e.g. [40]).At time step k the optimiser computes online an optimalopen loop sequence of control actions, so that the predictedoutputs follow a pre-specified reference signal and taking intoaccount possible hard and soft constraints. Only the currentcontrol action u(k|k) is actually fed to the plant over the timeinterval [k, k + 1). Next, at time step k + 1, the predictionand control horizons are shifted ahead by one step and a newopen loop optimisation problem is solved using the most recentmeasurements from the plant and the control action fed to theplant at previous discrete-time u(k|k).

Since the nonlinear model-based predictive control schemedeveloped in this work considers a local linear model of theplant derived at each discrete-time from the linearisation ofthe affine state-space neural network (Fig. 1), the remainingof this section will focus exclusively on the formulation ofthe constrained MPC tracking problem under the quadraticprogramming format.

Let the local linear discrete-time dynamic system be de-scribed in the state-space form as follows:

x (k + 1) = Ax (k) +B u (k)

y (k) = C x (k)(32)

with x ∈ Rn, y ∈ Rp, u ∈ Rm, A ∈ Rn×n, B ∈ Rn×m andC ∈ Rp×n. In addition, assume that the performance indexis defined as 2-norm and consider a set of linear constraintsimposed on the system’s inputs and outputs, along with abound on the rate of change of the control action vector. Underthe above premisses the open loop optimization problem canbe stated as follows:

∆U∗ = arg min∆U

P∑l=1

‖y (k + l|k)− r (k + l)‖2Ql

P−1∑l=1

‖u (k + l|k)‖2Rl+

M−1∑l=1

‖∆u (k + l|k)‖2Sl

(33)

subject to the system dynamics (32) and to the following setof constraints:

ymin ≤ y (k + l|k) ≤ ymax, l = 1, . . . , P, k ≥ 0

umin ≤ u (k + l|k) ≤ umax, l = 1, . . . , P − 1, k ≥ 0

|∆u (k + l|k)| ≤ ∆umax, l = 1, . . . ,M − 1, k ≥ 0

∆u (k + l|k) = 0 , l = M, . . . , P − 1, k ≥ 0

(34)

with Ql ∈ Rp×p, Rl, Sl ∈ Rm×p; ∆u ∈ Rm is the controlaction increment, r ∈ Rp denotes the reference signal and∆U∗ the extended optimal control action sequence.

Since the cost function is quadratic and the set of constraintsis linear, then optimization problem is convex and its solutionis unique. In this case, the constrained open loop optimalcontrol problem can be restated as a quadratic programming(QP) problem, namely,

∆U∗ = arg min∆U

hT∆U +

1

2∆UTH∆U

s.t. G ·∆U ≤ d

(35)

where G ∈ R(4m·M+2p·P )×m·M , d ∈ R(4m·M+2p·P ) and∆U ∈ Rm·M . The gradient h ∈ Rm·M and the HessianH ∈ Rm·M×m·M are given by:

hTj = 2

xT (k|k)

P−1∑i=j−1

(CAi+1

)TQi+1

i−j+1∑q=0

CAq

B−

P−1∑i=j−1

rT (i+ 1)Qi+1

i−j+1∑q=0

CAq

B+ uT (k − 1|k − 1)

P−1∑i=0

Ri

(36)

HTjj = 2

BT

P−j∑i=0

[i∑

q=0

(CAq)TQi+1

i∑q=0

CAq

]B

+

P−1∑i=j−1

Ri + Sj−1

(j = 1, . . . ,M)

(37)

6

HTjl = 2

BTP−1∑i=j−1

[i−j+1∑q=0

(CAq)TQi+1

i−l+1∑q=0

CAq

]B

+

M−1∑i=j−1

Ri

(j, l = 1, . . . ,M ; j 6= l)

(38)Concerning the constraint inequation, it follows that G is ablock matrix defined as:

G =

G1

G2

G3

G4

G5

G6

(39)

with

G1 = −G2

Im 0m · · · 0m 0m0m Im 0m 0m 0m

......

......

...0m 0m 0m 0m Im

(40)

G3 = −G4 =

Im 0m · · · 0m 0mIm Im 0m 0m 0m

......

......

...Im Im Im Im Im

(41)

G5 = −G6 =

φ11 0p×m · · · 0p×m 0p×mφ21 φ22 0p×m · · · 0p×m

......

......

...φM1 φM2 φM3 · · · φMP

......

......

...φP1 φP2 φP3 · · · φPM

(42)

where φij is given by:

φij = C

i−j∑q=0

Aq ·B, i = 1, . . . , P ; j = 1, . . . ,M∧j ≤ i (43)

With respect to the column vector d it takes the followingform:

d =

d1

d2

d3

d4

d5

d6

(44)

d1 = d2 =

∆umax (k|k)

∆umax (k + 1|k)...

∆umax (k +M − 1|k)

(45)

d3 =

umax (k + |k)− u (k − 1|k − 1)umax (k + 1|k)− u (k − 1|k − 1)

...umax (k +M |k)− u (k − 1|k − 1)

...umax (k + P − 1|k)− u (k − 1|k − 1)

(46)

d4 =

−umin (k + |k) + u (k − 1|k − 1)−umin (k + 1|k) + u (k − 1|k − 1)

...−umin (k +M |k) + u (k − 1|k − 1)

...−umin (k + P − 1|k) + u (k − 1|k − 1)

(47)

d5 =

cy1cy2...cyP

(48)

d6 =

cy

1

cy

2...cy

P

(49)

with

cyi = ymax−C

Aix (k|k) +

i−1∑j=0

AjBu (k − 1|k − 1)

(50)

and

cy

i = −ymin + C

Aix (k|k) +

i−1∑j=0

AjBu (k − 1|k − 1)

(51)

where ymax denotes the maximum allowable output from theplant, and ymin corresponds to the admissible minimum output.

IV. SOLAR POWER PLANT SETUP

A. Description

The distributed solar collector field (Acurex) used as atestbed for model-based predictive control experiments isowned by the Plataforma Solar de Almerıa (PSA), and islocated in the desert of Tabernas, in the South of Spain. TheAcurex plant is part of a small Solar Power System (SPS),which includes, in addition, a thermal storage tank system anda power unit (see Fig. 2).

The DSC field (Fig. 3) consists of 480 parabolic troughcollectors arranged in 20 rows aligned on a West-East axisand forming 10 independent loops. Each solar collector is alinear parabolic-shaped reflector that focuses the sun’s beamradiation on a linear absorber pipe located at the focus of theparabola. Each loop is 172 m long, with an active sectionof 142 m, while the reflective area of the mirrors is around264.4 m2.

The heat transfer fluid used to transport the thermal energyis the Santotherm 55, which is a synthetic oil with a maximum

7

loop 10

Pump

Steam generator/desalination

plant

Storage

Tank

rrI

outT

inT

loop 2

loop 1

Fig. 2. PSA’s Solar Power System schematics.

Fig. 3. Acurex solar collector loop.

film temperature of 318 C and an autoignition temperatureof 357 C. The thermal oil is heated as it circulates throughthe absorber pipe before entering the top of the storage tank.The colder inlet oil is extracted form the bottom of the tank,while a three way valve located at the field outlet enablesthe oil recycling (by-passing the storage tank) until the outletoil temperature is high enough to be sent to the tank. Thethermal energy stored in the tank is subsequently used togenerate electricity in a conventional steam turbine/generatoror consumed in the solar desalination plant. The main reasonfor including a storage tank in the SPS is to provide abuffer between the electricity production and the solar energy

availability.The DSC field is provided with a sun trackingsystem, which causes the solar collector to revolve aroundan axis parallel to the receiver in order to follow the yearlyvariation of the sun’s declination.

The DSC field operating limits are, in terms of oil flowrate, restricted to a minimum of 2.0 l/s and a maximum of12.0 l/s. The lower bound is imposed to reduce the risk ofoil decomposition, which occurs when its temperature exceeds305 C, while the maximum bound is related to the admissiblepressure drop along the pipe length.

From the control point of view, the main difference betweena standard power plant and a thermal solar power plant isthat the energy source (solar radiation) cannot be manipulated.Moreover, the direct solar radiation is seasonally dependent,varies throughout the day and is susceptible to atmosphericconditions, such as cloud cover, humidity and turbidity. Inthis context, the objective of the control system is to maintainthe outlet oil temperature at a prescribed value irrespectiveof variations on the beam solar radiation, inlet oil temperatureand solar collectors’ reflectivity, while taking into account hardconstraints imposed on the system, and by manipulating theoil flow rate through each loop.

B. Solar Collector Field Identification

Regarding the identification of the Acurex field for controlpurpose, the outlet oil temperature Tout (system output) de-pends not only on the oil flow rate Qin(input), but also on othervariables, such as the inlet oil temperature Tin, solar radiationIrr, ambient temperature and mirrors reflectivity. Since thesolar beam radiation and the inlet oil temperature are mea-surable, it is worth incorporating a feedforward compensator(Fig. 4) in series with the plant (see e.g. [4]).

Compensator ACUREX

Tin

Tref

Irr

Qin Tout

Fig. 4. Feedforward compensator.

This feedforward term is derived for steady state conditionsand delivers the required oil flow rate demanded to the pump,provided the desired outlet oil temperature Tref, solar beamradiation and inlet oil temperature are known. The feedfor-ward term was implemented according to following generalexpression:

Qin =µ ·Nact · Ic ·Aeff

ρ · cp (Tref − Tin)(52)

where µ is a coefficient to be computed experimentally, Nactthe number of active loops, Ic the corrected solar radiationcomputed from the solar beam radiation Irr, Aeff denotes theeffective mirrors’ surface of each loop, ρ is the oil specificmass and cp is the specific thermal capacity.

The input to the feedforward compensator is the referenceoutlet oil temperature, while the corresponding output isassociated with the oil flow rate, and the remaining variables,

8

0

50

100

150

200

250

300

10 11 12 13 14 15 16

0

2

4

6

8

10

Local Time [h]

Tout

Q in

Tem

per

atu

re [

C]

Flo

w r

ate

[l/

s]

700

750

800

850

900

950

1000

10 11 12 13 14 15 16

50

100

150

200

250

300

Local Time [h]

Irr

Tin

Radia

tion [W

/m

2]

Tem

per

atu

re [

C]

Fig. 5. Data set collected from the Acurex field.

namely, Tin and Ic, can be regarded as measurable distur-bances acting on the pseudo-system.

For the Acurex identification two data sets have been col-lected from the plant to be used in the neural network trainingand validation phases. They included the outlet oil temperature(Tout), the inlet oil temperature (Tin), the flow rate (Qin) andthe solar beam radiation (Irr), at a sample rate of 15 s. Oneof those data sets is plotted in Fig. 5. In order to come with anestimate for the effective order of the affine state-space neuralnetwork the approach proposed in Sec. II-B is then applied.Taking into account the data set presented in Fig. 5 and anumber of row blocks chosen as i = 20, which is substantiallyhigher than the expected effective dominant singular valuescardinality, the singular values associated with the projectionmatrix Π20 are presented in (53). From the singular valuesinspection, in terms of absolute magnitude and magnitudedecay, it is clear that the dominant vector basis is quite lowerthan the rank of Π20. Hence, a post-processing procedureshould be considered to reduce the model’s complexity. Itshould be mentioned that, in the present case, full ranknessis mainly due to nonlinear dynamics embedded in the datasets, and also to a finite data set (j << ∞) collected fromthe plant.

In order to reduce the vector basis dimension let us consider

the methodology based on (10), that is,

r = max r : log σr > −0.1920

The largest integer satisfying the previous inequality is 6,which points to a 6th order model to approximate the plant’sdynamics embedded in the collected data and considering alinear approximation under the form of Eq. (2).

σ =

1381.62381428.65087527.8721505.3127261.8146151.0548680.7196110.3897580.3872680.3546880.2936510.2790580.2626570.1659070.1381880.0999480.0991640.0383630.0315530.000493

(53)

To assess the 6th order affine state-space model perfor-mance, the neural network was trained offline using theLevenberg-Marquardt algorithm, within the MATLAB c© envi-ronment. Fig. 6 compares the outlet oil temperature collectedon the Acurex field with the corresponding prediction pro-vided by the affine state-space neural network. As can beobserved, the neural predictor’s output is in line with thedistributed solar collector field behaviour, either in terms ofovershoot, settling time and static gain. Nevertheless, a slightmodel-plant mismatch is still detected, which is confirmedby the Mean Normalized Absolute Error (MNAE) magnitude,namely, 1.10 × 10−2. This fact suggests the need for animprovement in terms of generalization capabilities of theneural network approximator, by considering, for instance, arecursive-based updating approach.

Sample

Tem

per

atu

re [ºC

]

200 300 400 500 600 700 800 900 1000 11000

50

100

150

200

250

300

Acurex

Simulation

Fig. 6. Acurex outlet oil temperature versus neural network simulation.

9

V. SOLAR COLLECTOR FIELD TESTS

The nonlinear MPC scheme was tested on the Acurexfield of PSA in two different strategies, namely, indirectadaptive and non-adaptive. In either cases, for regulating ordriving the outlet oil temperature to a pre-specified level,despite variations in the sun’s beam radiation and in inlet oiltemperature, the control system manipulates the thermal oilflow rate pumped to the solar collector field.

Given the computation complexity and memory require-ments of this approach, it was chosen to run the main routines,namely, online identification, in the case of adaptive schemes,state estimation, and the open loop optimisation, on a remotecomputer, more specifically a laptop computer. This allowed toimplement these routines in MATLAB c© taking advantage ofits programming flexibility and the available toolboxes, suchas the optimisation toolbox, for solving the required open loopconstrained quadratic optimisation problem.

LaptopTerminal

Serial Cable

TX/RX

Server

Fig. 7. Laptop-Server serial communication.

The laptop computer was connected with the Acurex fieldServer via RS232 serial communication (Fig. 7), which pro-vides the physical means for the exchange of data. In thisframework, the Acurex control server supplies the laptop withthe DSC field data, which computes a new control actionsequence under the form of oil flow rate. The new oil flowrate u(k|k) is then sent to the laptop’s COM and read bythe Acurex field server. Next, the Server forwards the controlaction, regarded as a reference flow rate, to the pump PI(Proportional-Integral) controller. The communication routinesfor synchronisation, sending and reading have been imple-mented both in C code and included in the Acurex softwarepackage and also in MATLAB c© to be called within the controlcycle implemented on the remote laptop.

In all the experiments reported in this work it was con-sidered a prediction horizon P = 10 time-steps, a controlhorizon M = 1 and the weight matrices associated with theperformance index presented in (54). These penalty matriceswere selected by a trial and error heuristic, using a simulationsoftware package for the Acurex Solar collector field [41],and considering as design-based conditions the step response’sovershoot and settling-time.

Qi =

5, i = 1, . . . , P − 1

100, i = P

Ri = 10−3, i = 1, . . . , P − 1

Si = 10−4, i = 1, . . . ,M − 1

(54)

In what the hard constraints included in the optimizationproblem are concerned, they reflect physical limitations of theAcurex field. The oil flow rate was set as lying within 2.0 l/sand 10.0 l/s, being the upper bound lower than the maximumadmissible flow rate for safety reasons, while the outlet oiltemperature is limited to 300 C. In addition, the maximumoil flow rate increments were constrained to 0.1 l/s, in orderto enable a smooth pump operation, which contributes tomaximizing the main oil pump lifespan. In terms of samplingtime, unlike the experiment reported in Section V-C, in whichwas considered a sampling time of 10 s, all the remainingexperiments were conducted considering a sampling time of15 s.

A. Non-Adaptive MPC

This experiment concerns an MPC control configuration inwhich the neural affine state-space neural network predictoris not updated in the course of the conducted tests, and theunscented Kalman filter is only used for state estimation .

The main purpose of this experiment is twofold: to evaluatethe performance of a non-adaptive framework in controllingthe Acurex field and to use it on comparison basis to assessimproved MPC schemes performance, namely the indirectadaptive approach.

Tem

per

atu

re [°C

]

11 12 13 14 150

50

100

150

200

250

300

160

2

4

6

8

10

Flo

w R

ate

[l/

s]

Local Time [h]

ref

out

in

Tem

per

atu

re [°C

]

Sola

r R

adia

tion [W

/m

2]

11 12 13 14 15100

150

200

250

300

16500

600

700

800

900

1000

Local Time [h]

rr

in

Fig. 8. Non-adaptive MPC.

10

In Fig. 8 are presented the outcomes of one of the exper-iments carried out at the site, for which the nonlinear modelparameters are fixed, and assuming Px(0) = I, see Eq. (24).In this figure it is shown the outlet oil temperature (Tout), theoil reference temperature (Tref ), the control action (u(k|k)),under the form of demanded flow rate in each loop (Qin), theoil inlet temperature (Tin) and the solar beam radiation priorto correction (Irr), over time.

As can be observed from Fig. 8, the pump has a smoothoperation, although a residual static offset is evident for someoperating point, which is mainly attributed to model-plantmismatch. Moreover, the control system performance is notsignificantly affected by disturbances on solar beam radiationand inlet oil temperature, as they are accommodated by thefeedforward compensator expressed in terms of Eq. (52).

B. Indirect Adaptive MPC

This Section considers the implementation of an indirectadaptive MPC scheme on the Acurex field, for which theplant parameters are adjusted online in order to cope withunmodelled dynamics and disturbances. A dual unscentedKalman filter, as proposed in Section II-C, was used forrecursive parameter estimation and state estimation. Initialcovariance matrices were chosen as Pw(0) = 5.0 × 10−3Iand Px(0) = I. Additionally, a forgetting factor τ = 0.9986was also considered by rewriting Eq. (14) as:

Pw (k|k − 1) =1

τPw (k − 1|k − 1) (55)

The Fig. 9 shows results of an experiment where theadaptive MPC scheme was incorporated in the control loop.From these results it is possible to infer that the controlsystem is “well-behaved” despite disturbances on the solarbeam radiation and inlet oil temperature. Furthermore, giventhe indirect adaptive trait of this controller, which relies onrecursive nonlinear identification techniques, modelling errorsare progressively attenuated, which benefits the steady-statecontrol system behaviour, namely the static error, in compari-son with non-adaptive schemes. Finally, a mention should bemade to the fact that right after the control policy has beenswitched to the MPC, a rather prominent overshoot is noticed,along with slight oscillations in the outlet oil temperature,which can be explained in part by the adaptation mechanismsthat are in play.

C. Indirect Adaptive MPC - Actuation Fault

This experiment intends to assess the control system be-haviour and robustness for an operation fault on the actuationsystem. This fault was taking place on the variable speed driveattached to the main pump, and was imposing a differencein the pumped oil flow rate with respect to the referenceoil flow rate provided by the MPC scheme. In the case ofthe experiment in question, the flow rate deviation given bythe difference between the prescribed oil flow rate and themeasured value in loco is presented in Fig. 10. As can beobserved, most of the time the prescribed oil flow rate was

10 11 12 13 14 15 160

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300

10 11 12 13 14 15 160

2

4

6

8

10

Tem

per

atu

re [°C

]

Flo

w R

ate

[l/

s]

Local Time [h]

ref

out

in

10 11 12 13 14 15 16100

150

200

250

300

500

600

700

800

900

1000

Tem

per

atu

re [°C

]

Local Time [h]

Sola

r R

adia

tion [W

/m

2]

rr

in

Fig. 9. Indirect adaptive MPC.

higher than that fed to the Acurex field. As a result, this willlead to a positive offset of the oil flow rate temperature ifthe corresponding effect is not appropriately accommodatedby the control system.

The experiment was run using the indirect adaptive MPCscheme with a sampling time of 10 s, and considering the samecovariance matrices as those presented in Sec. V-B, togetherwith τ = 0.9987. The obtained results are shown in Fig. 11.

As can be observed from this figure, the indirect adaptiveMPC scheme is able to drive the outlet oil temperature to thecorresponding reference, in spite of a malfunctioning pendingon the variable speed drive. The accommodation of this faultis carried out by the joint contribution of online identifica-tion, which incorporates into the Acurex field dynamics theactuation fault, and the state estimation procedure.

In what the sampling time is concerned, no significant dif-ference is detected in terms of performance by choosing morestringent admissible values for the sampling time. However,further studies have to be conducted on site to confirm thisassertion.

VI. CONCLUSIONS

This paper presented the implementation of a constrainedaffine state-space neural network based predictive control

11

Local Time [h]

11 12 13 14 15 16-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Flo

w R

ate

Dev

iati

on [l/

s]

Fig. 10. Oil flow rate deviation.

Tem

per

atu

re [°C

]

Flo

w R

ate

[l/

s]

Local Time [h]

ref

out

in

11 12 13 14 15 160

50

100

150

200

250

300

0

2

4

6

8

10

Tem

per

atu

re [°C

]

Local Time [h]

Sola

r R

adia

tion [W

/m

2]rr

in

11 12 13 14 15 16100

150

200

250

300

500

600

700

800

900

1000

Fig. 11. Indirect adaptive MPC - actuation fault.

scheme on a distributed solar collector (Acurex) field, ownedby the Plataforma Solar de Almerıa, where the main goalconsists in driving the outlet oil temperature to a prescribedvalue. The neural network predictor’s complexity measuredby the order of the state-space nonlinear model, is dealt withby estimating the number of dominant singular values of asubspace oblique projection of data driven Hankel matrices.

In order to guarantee that the optimization problem convexityis held, the dynamic model of the Acurex field, described by anaffine state-space neural network, is locally linearised, whilethe optimisation problem constraints are represented by a set oflinear inequalities. The online updating of the neural networkweights and state estimation is based on a dual unscentedKalman filter. In the case of the non-adaptive MPC frameworkonly a state estimation is carried out.

Experiments conducted on the Acurex field included testsbased on a non-adaptive MPC scheme and on an indirectadaptive version. The first implementation showed that asa result of model-plant mismatch the outlet oil temperaturedisplays a static offset, irrespective of the operating regime.This effect was practically removed in the indirect-adaptiveformulation, due to a neural model improvement, which isgradually achieved by means of a recursive adjusting ofweights, based on new data collected from the plant.

Concerning the MPC robustness to measurable disturbances,such as solar beam radiation and inlet oil temperature, theywere effectively accommodated by the feedforward compen-sator as the experiments reveal. In addition, in the case of afaulty actuator, resulting from a malfunction on the variablespeed drive that is attached to the main pump of the Acurexfield, the indirect adaptive MPC framework was able toaccommodate its impact on the outlet oil temperature.

ACKNOWLEDGMENT

The experiments included in this work have been carried outwithin the project Improving Human Potential Program (EC-DGXII), supported by the European Union Program Trainingand Mobility of Researchers. The authors would like to expresstheir gratitude to the staff of the Plataforma Solar de Almerıa.

REFERENCES

[1] K. Lei and S. Zhou, “Per capita resource consumption and resourcecarrying capacity: A comparison of the sustainability of 17 mainstreamcountries,” Energy Policy, vol. 42, no. 0, pp. 603 – 612, 2012.

[2] S. V. A. R. Sastry and P. Sreenu, “New energy sources and its sus-tainability,” in Engineering Education: Innovative Practices and FutureTrends (AICERA), 2012 IEEE International Conference on, July 2012,pp. 1–15.

[3] G. Manzolini, M. Bellarmino, E. Macchi, and P. Silva, “Solar thermody-namic plants for cogenerative industrial applications in southern europe,”Renewable Energy, vol. 36, no. 1, pp. 235 – 243, 2011.

[4] E. Camacho, F. Rubio, M. Berenguel, and L. Valenzuela, “A survey oncontrol schemes for distributed solar collector fields. part I: Modelingand basic control approaches,” Solar Energy, vol. 81, no. 10, pp. 1240– 1251, 2007.

[5] ——, “A survey on control schemes for distributed solar collector fields.part II: Advanced control approaches,” Solar Energy, vol. 81, no. 10,pp. 1252 – 1272, 2007.

[6] E. F. Camacho and M. Berenguel, “Robust adaptive model predictivecontrol of a solar plant with bounded uncertainties,” InternationalJournal of Adaptive Control and Signal Processing, vol. 11, no. 4, pp.311–325, 1997.

[7] C. Greco, G. Menga, E. Mosca, and G. Zappa, “Performance improve-ments of self-tuning controllers by multistep horizons: The musmarapproach,” Automatica, vol. 20, no. 5, pp. 681 – 699, 1984.

[8] F. Coito, J. M. Lemos, R. N. Silva, and E. Mosca, “Adaptive controlof a solr energy plant: Exploiting accessible disturbances,” InternationalJournal of Adaptive Control and Signal Processing, vol. 11, no. 4, pp.327–342, 1997.

[9] E. Camacho, Advances in Model-Based Predictive Control. OxfordUniversity Press, 1994, ch. Application of GPC to a solar power plant.

12

[10] R. Silva, L. Rato, J. Lemos, and F. Coito, “Cascade control of adistributed collector solar field,” Journal of Process Control, vol. 7, no. 2,pp. 111 – 117, 1997.

[11] R. Silva, N. Filatov, J. Lemos, and H. Unbehauen, “Feed-back/feedforward dual adaptive control of a solar collector field,” inControl Applications, 1998. Proceedings of the 1998 IEEE InternationalConference on, vol. 1, Sep 1998, pp. 309 –313 vol.1.

[12] R. Pickhardt and R. Neves da Silva, “Application of a nonlinearpredictive controller to a solar power plant,” in Control Applications,1998. Proceedings of the 1998 IEEE International Conference on, vol. 1,Sep 1998, pp. 6 –10 vol.1.

[13] K. Patan, “Stability analysis and the stabilization of a class of discrete-time dynamic neural networks,” Neural Networks, IEEE Transactionson, vol. 18, no. 3, pp. 660 –673, May 2007.

[14] X. Han, W.-F. Xie, Z. Fu, and W. Luo, “Nonlinear systems identificationusing dynamic multi-time scale neural networks,” Neurocomputing,vol. 74, no. 17, pp. 3428 – 3439, 2011.

[15] H. Wei, S. Billings, Y. Zhao, and L. Guo, “An adaptive wavelet neuralnetwork for spatio-temporal system identification,” Neural Networks,vol. 23, no. 10, pp. 1286 – 1299, 2010.

[16] B. S. Leon, A. Y. Alanis, E. N. Sanchez, E. Ruiz-Velazquez, andF. Ornelas-Tellez, “Inverse optimal neural control for a class of discrete-time nonlinear positive systems,” International Journal of AdaptiveControl and Signal Processing, 2012.

[17] J. Henriques, P. Gil, P. Carvalho, H. Duarte-Ramos, and A. Dourado,Power Plant Applications of Advanced Control Techniques. ProcessEngEngineering GmbH, 2010, ch. Nonlinear Control Based on Affine NeuralNetworks: Applicatiion to a Solar Power Plant, pp. 295–320.

[18] S. Julier, J. Uhlmann, and H. Durrant-Whyte, “A new method forthe nonlinear transformation of means and covariances in filters andestimators,” Automatic Control, IEEE Transactions on, vol. 45, no. 3,pp. 477 –482, mar 2000.

[19] A. Juditsky, H. Hjalmarsson, A. Benveniste, B. Deylon, L. Ljung,J. Sjberg, and Q. Zhang, “Nonlinear black-box modeling in systemidentification: Mathematical foundations,” Automatica, vol. 31, pp. 1725– 1750, 1995.

[20] J. Sjberg, Q. Zhang, L. Ljung, A. Benveniste, B. Deylon, P. yvesGlorennec, H. Hjalmarsson, and A. Juditsky, “Nonlinear black-boxmodeling in system identification: a unified overview,” Automatica,vol. 31, pp. 1691–1724, 1995.

[21] G. Cybenko, “Approximation by superpositions of a sigmoidal function,”Math. Control, Signals Syst, vol. 2, pp. 303–314, 1989.

[22] K. Funahashi, “On the approximate realization of continuous mappingsby neural networks,” Neural Networks, vol. 2, pp. 183–192, 1989.

[23] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforwardnetworks are universal approximators,” Neural Networks, vol. 2, pp.359–366, 1989.

[24] M. Leshno, V. Lin, A. Pinkus, and S. Schocken, “Multilayer feedforwardnetworks with a nonpolynomial activation function can approximate anyfunction,” Neural Networks, vol. 6, no. 6, pp. 861–867, 1993.

[25] Siegelmann, Horne, and Giles, “Computational capabilities of recurrentNARX neural networks,” IEEETSMC: IEEE Transactions on Systems,Man, and Cybernetics, vol. 27, no. 2, pp. 208–215, 1997.

[26] E. Sontag, “Recurrent neural networks: Some systems-theoretic aspects,”in Dealing with Complexity: a Neural Network Approach. Springer-Verlag, 1997, pp. 1–12.

[27] D. T. Pham and X. Liu, Neural Networks for Identification, Predictionand Control. Springer, 1995.

[28] Q. Dong, K. Matsui, and X. Huang, “Existence and stability of periodicsolutions for hopfield neural network equations with periodic input,”Nonlinear Analysis: Theory, Methods & Applications, vol. 49, no. 4, p.471 479, 2002.

[29] W. Duch and N. Jankowski, “Survey of neural transfer functions,” NeuralComputing Surveys, vol. 2, pp. 163–212, 1999.

[30] P. Gil, J. Henriques, A. Cardoso, and A. Dourado, “On affine state-spaceneural networks for system identification: Global stability conditions andcomplexity management,” Control Engineering Practice, vol. 21, no. 4,pp. 518 – 529, 2013.

[31] P. Gil, J. Henriques, A. Dourado, and H. Duarte-Ramos, “Order es-timation in affine state-space neural networks,” in Soft Computing inIndustrial Applications, 2005. SMCia/05. Proceedings of the 2005 IEEEMid-Summer Workshop on, June 2005, pp. 132 – 137.

[32] V. Sima, D. M. Sima, and S. V. Huffel, “High-performance numericalalgorithms and software for subspace-based linear multivariable systemidentification,” Journal of Computational and Applied Mathematics, vol.170, no. 2, p. 371397, 2004.

[33] B. D. Moor, P. Overschee, and W. Favoreel, “Algorithms for subspacestate-space identification: An overview,” in Applied and computationalcontrol, signals, and circuits, B. N. Datta, Ed. Birkhuser, 1999, vol. 1,ch. 6, pp. 247 – 311.

[34] K. Astrom, “Maximum likelihood and prediction error methods,” Auto-matica, vol. 16, no. 5, p. 551574, 1980.

[35] L. Ljung, System Identification Toolbox: Users’s Guide. The Math-Works, Inc., 1995.

[36] H. Yin, Z. Zhu, and F. Ding, “Model order determination using thehankel matrix of impulse responses,” Applied Mathematics Letters, Jan.2011.

[37] P. Gill and W. Murray, “Algorithms for the solution of the nonlinearleast-squares problem,” SIAM Journal on Numerical Analysis, vol. 15,no. 5, pp. 977–992, 1978.

[38] A. Suratgar, M. Tavakoli, and A. Hoseinabadi, “Modified levenberg-marquardt method for neural networks training,” in Proceedings - Wec05: Fourth World Enformatika Conference, 2005.

[39] E. Wan and R. Van Der Merwe, “The unscented kalman filter fornonlinear estimation,” in Adaptive Systems for Signal Processing, Com-munications, and Control Symposium 2000. AS-SPCC. The IEEE 2000,2000, pp. 153 –158.

[40] J. Maciejowski, Predictive Control with Constraints. Prentice Hall,2002.

[41] M. Berenguel, E. Camacho, and F. Rubio, “Simulation software pack-age for the acurex field,” Departamento de Ingenierıa de Sistemas yAutomatica, ESII, Universidad de Sevilla, Tech. Rep., 1994.